Response to comment on “Heterodyne lidar returnsin the turbulent atmosphere: performance evaluationof simulated systems” by Frehlich and Kavaya

Aniceto Belmonte and Barry J. Rye

The differences in approach and misunderstandings that account for the problems described in Frehlichand Kavaya’s comment �Appl. Opt. 41, 1595 �2002�� are summarized. We also acknowledge an omissionin our paper that has been drawn to our attention separately. © 2002 Optical Society of America

OCIS codes: 010.3640, 010.7060, 010.1330.

When we started using numerical simulations tostudy the effects of refractive turbulence on hetero-dyne lidar systems, to our knowledge no similar stud-ies existed in the literature. Although the subjectwas first discussed over 20 years ago, the short list ofreferences �Refs. 1–5� listed in Frehlich and Kavaya’scomment1 �FK02� indicates that the available theoryis somewhat sparse �and experimental data are evenmore so�. Published theory concentrates on some-what simplified situations, whereas we believed thatsimulation has the potential for greater realism.Our intention was not to comment on fundamentaltheory, least of all path-integral theory, although weare somewhat flattered that anyone might think wedid. We did want to compare results from existingtheory with realistic simulations that were achiev-able using readily available computer resources.We were concerned more with the monostatic geom-etry commonly used in lidar than with the idealizedbistatic geometry, which is theoretically more tracta-ble. Proper understanding of monostatic lidarsdates from the First Coherent Laser Radar Confer-ence in 1980 and, in particular, from two publishedarticles that followed papers at that meeting. One ofus2 used a two-beam �transmitted and backpropa-

A. Belmonte �belmonte@tsc.upc.es� is with the Department ofSignal Theory and Communications, Universitat Politecnica deCatalunya, 08034 Barcelona, Spain. B. J. Rye �barry.rye@noaa.gov� is with the Cooperative Institute for Research in Envi-ronmental Science, NOAA�OAR�ETLR�ET2, 325 Broadway,Boulder, Colorado 80305-3328.

Received 19 October 2001; revised manuscript received 20 No-vember 2001.

0003-6935�02�091601-03$15.00�0© 2002 Optical Society of America

gated or phase-conjugate local oscillator� approach topoint out, contrary to the conventional wisdom of thetime, that returns from monostatic lidars should beenhanced relative to idealized bistatic lidars andthat physically the enhancement was rather simplyrelated to target-plane scintillation; Clifford andWandzura3 �CW81� gave analytic expressions formonostatic lidar returns. Together, these papers in-dicated that returns should also be enhanced abso-lutely, i.e., compared with lidar returns with norefractive turbulence.2 This conclusion was alsothat of Murty,4 who in the weak turbulence limit wasable to separate analytically the contributions ofbeam truncation, coherence loss due to beam spread-ing, and scintillation gain in monostatic systems. Inthe paper by Belmonte and Rye5 �BR00� that haselicited the comment in FK02, we used Frehlich andKavaya’s6 �FK91� more recent paper rather thanCW81 because it provides approximate formulas thatare easily used and deals with the subject at consid-erably greater length. The earliest description inthe public domain of our research7 was submitted forthe Tenth Coherent Laser Radar Conference in 1999where, together with the paper of Banakh et al.8 �alsosubsequently published9�, it introduced the hetero-dyne lidar community to the potential of simulationin this field.

The principal contention in FK02 �abstract� is thatwe state that their series path-integral expansionwas developed under a square-law structure-functionapproximation. Nowhere did we state this, and wedid not intend to imply it. In preparing our Figs.16–21,5 we used equations from FK91, which are saidto be based on “the use of . . . the square-law approx-imation for the structure function.” This or a simi-lar phrase appears in Section 4 of FK91, “Gaussian

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Lidar System,” prior to each of Eqs. �178�, �204�,�210�, and �214�. We assumed �and still assume�that it refers to an approximation discussed earlier�for example, by Yura10�, in which the 5�3 exponent isreplaced by 2 in an equation for the transverse co-herence length. The approximation is generally re-garded as useful to facilitate recovery of analyticresults for the idealized bistatic geometry in whichthe two beams are supposed to pass through entirelyseparate atmospheric paths. We use the expressionquadratic or square-law structure function in thissense. Indeed, it is not unlikely that the phrase wasused in this same sense by Clifford and Wandzura,for example, when they referenced earlier lidar re-ports in CW81. Our use of the phrase has nothing todo with path-integral theory. We understand thatuse of the same terminology in two different contextscould be misleading.

With regard to the results, the two-beam approachof Rye2 has been used to generate the simulations inBR00, FK91, and FK02. The differences amongthese papers relate to goals, choice of beam profile,and outer scale. In BR00, our simulation data werecompared with the FK91 model essentially to see ifthe latter gave a usable approximation for realisticsystems. We assumed a lidar using a Gaussianbeam truncated by circular apertures, with a finiteouter turbulence scale L0. In FK91, the aperture ismodeled by use of a Gaussian pupil function to trun-cate a Gaussian beam. Theoretically, this is an un-necessary complication because, when a Gaussianpupil function is multiplied by a Gaussian beam pro-file, it just leads to a smaller Gaussian beam. Be-cause the intention appeared to be to model a realisticaperture model, we retained it in our comparisons.The Gaussian pupil model is not used in FK02. Inboth FK91 and FK02, a Kolmogorov turbulence spec-trum with infinite outer scale is assumed.

The infinite outer scale is simulated in FK02 by useof a standard random subharmonic method. As theauthors show, this is desirable when an exact com-parison is made with traditional theory because theassumed infinite outer scale cannot be simulatedwith a discrete grid. In the discussion of results foran idealized bistatic system, it is implied in Sections4 and 5 of FK02 that we also intended to use a Kol-mogorov spectrum but that an outer scale was “im-posed”1 on us in an apparently arbitrary way by thedimensions of the simulation. However, as clearlystated in BR00, we used a von Karman spectrum,which presupposes the definition of a finite outerscale.

The choice of both inner and outer scale with rela-tion to the grid size was discussed in great detail byone of us in a preparatory paper,11 which representsa serious attempt to define the simulation parame-ters in a realistic way. At low altitude h, the turbu-lence spectrum depends on the topography of theEarth’s surface and local meteorological conditions.For example, it is often assumed that L0 � 2�5 h,which predicts outer scales as large as several tens ofmeters even in surface-layer turbulence. However,

measurements suggest a different dependence ofouter scale on altitude, and several empirical modelsrestrict the outer scale to less than five meters.12

This corresponded roughly to the size of the numer-ical grid assumed in our simulations, i.e., the limitingscales determined the grid and not vice versa. Thedifference between reality and traditional theory re-garding scale sizes had been recognized in CW81 withthe comment that their calculation implicitly takesthe outer scale to be infinite whereas �in practice� theouter scale is typically of the order of 100 cm. Theeffect of the outer scale is more pronounced in bistaticthan in monostatic systems because of the latter’simmunity to beam wander or tilt.3

The explanation proposed in FK02 �Section 4� thatthe departure from theory of the simulation results inBelmonte’s11 Fig. 7 arises because large-scale tilts arenot correctly represented is incorrect. It ignores theexplanation originally given11; that is, that the beambreaks up so that the basic assumption of a Gaussianbeam no longer applies. One advantage of the beamsimulation is that the structure of the beam can beexamined at any point, which in this case allowed theoriginal explanation to be verified readily. When re-alistic outer scales are considered under the strongturbulence conditions of Fig. 7, little of the long-termaveraged beam spread is due to beam wander. Theissues of beam spread and wander are discussed fur-ther in Ref. 11 in relation to Figs. 8–12.

The Hill spectrum used in FK02 describes the at-mospheric turbulence over a wide range of spatialscales and is frequently chosen for its mathematicalutility and simplicity. In particular, it features abump at high wave numbers near 1�l0, where l0 is theturbulent inner scale. Andrews13 has developed acomprehensive analytical approximation to the Hillspectrum that includes an outer-scale parameter andallows proper consideration of turbulence scales overall ranges in the simulations. At low wave numbers�k � 1�L0�, the reduction in values of the spectrumcaused by the presence of finite outer scale is similarin both this approach to the Hill spectrum and thevon Karman spectrum used in our simulations. As ameans of including in the simulations the experimen-tally observed high-wave-number bump, use of An-drews’s approach to the Hill spectrum was discussedand anticipated as a future development in Ref. 11.

Use of the Hill spectrum or approximations to itwill tend to increase the strength of target-plane scin-tillation. Because this accounts for return signal en-hancement in monostatic systems,2 the results inBR00 for the monostatic geometry can be expected tobe somewhat conservative. Even so, our main con-clusion was that simulations indicate significantlygreater enhancement of the lidar return than ispredicted in the formula provided by FK91. Wecommented that the problem may lie not with theirzero-order approximation to the path-integral seriesexpansions �and certainly not with any supposed useof a square-law structure function�. Instead, wequestioned use of the weighting term chosen reason-ably, but apparently arbitrarily, to combine low- and

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high-frequency terms and to provide an analytic re-sult at intermediate levels of turbulence where theeffects of scintillation are often of most interest. Useof this weighting term was described in FK91 as heu-ristic. The graph in Fig. 2 of FK02 also shows en-hancement significantly greater than that given byFK91’s heuristic model. It appears that there is nofundamental difference between our results for thiscase.

There is an omission in our paper that has beendrawn to our attention privately by our Russian col-leagues, and we should like to take this opportunityto share this observation. Enhanced backscatter14

resulting from propagation through turbulent mediawas extensively analyzed in the Russian literatureprior to 1980. English language review articleswere later published to present this research to awider audience.15,16 The relevance of the Russianresearch to heterodyne lidar is perhaps not immedi-ately obvious. The introductory discussions in thisliterature often concern a point source illuminating apoint target rather than engineering-oriented studiesof beams and extended targets. Optically coherentsystems are necessarily considered, but there is nospecific mention of heterodyne receivers, so that thedevelopment brings to mind use of diffraction-limiteddirect detection systems or optical homodyning.However, all the essential physics of the problem arecontained in their analysis. With proper awarenessof this research, the insights obtained in the hetero-dyne lidar literature in 1980 and 1981 would proba-bly have been anticipated. With regard to BR00, welisted the enhancement in incoherent backscatterfrom far-field targets in weak turbulence as a novelresult. We should have acknowledged that thisfinding merely confirms what had been predicted17,18

in or before 1973 and, incidentally, was used at thattime to postulate absolute enhancement.

References and Notes1. R. Frehlich and M. J. Kavaya, “Comment on ‘Heterodyne lidar

returns in the turbulent atmosphere: performance evalua-tion of simulated systems’,” Appl. Opt. 41, 1595–1600 �2002�.

2. B. J. Rye, “Refractive turbulence contribution to incoherentbackscatter heterodyne lidar returns,” J. Opt. Soc. Am. 71,687–691 �1981�.

3. S. F. Clifford and S. Wandzura, “Monostatic heterodyne lidarperformance: the effect of the turbulent atmosphere,” Appl.Opt. 20, 514–516 �1981�.

4. R. Murty, “Refractive turbulence effects on truncated Gauss-ian beam heterodyne lidar,” Appl. Opt. 23, 2498–2502 �1984�.

5. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in the

turbulent atmosphere: performance evaluation of simulatedsystems,” Appl. Opt. 39, 2401–2411 �2000�.

6. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar per-formance for general atmospheric refractive turbulence,” Appl.Opt. 30, 5325–5352 �1991�.

7. A. Belmonte, B. J. Rye, W. A. Brewer, and R. M. Hardesty,“Coherent lidar returns in turbulent atmosphere from simu-lation of beam propagation,” in Proceedings of the Tenth Bien-nial Coherent Laser Radar Conference, Mt. Hood, Oregon, 28June–2 July 1999 �Universities Space Research Association,Huntsville, Ala., 1999�, pp. 78–81.

8. V. A. Banakh, I. N. Smalikho, and C. Werner, “Effect of refrac-tive turbulence on Doppler lidar operation in atmosphere:numerical simulation,” in Proceedings of the Tenth BiennialCoherent Laser Radar Conference, Mt. Hood, Oregon, 28June–2 July 1999 �Universities Space Research Association,Huntsville, Ala., 1999�, pp. 82–85.

9. V. A. Banakh, I. N. Smalikho, and C. Werner, “Numericalsimulation of the effect of refractive turbulence on coherentlidar return statistics in the atmosphere,” Appl. Opt. 39, 5403–5414 �2000�.

10. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systemsin the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 �1979�; also in Surveillance of Environmental Pollutionand Resources by Electromagnetic Waves, T. Lund, ed. �Reidel,London, 1978�, pp. 67–93.

11. A. Belmonte, “Feasibility study for the simulation of beampropagation: consideration of coherent lidar performance,”Appl. Opt. 39, 5426–5445 �2000�.

12. C. E. Coulman, J. Vernin, Y. Coqueugniot, and J. L. Caccia,“Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 �1988�.

13. L. C. Andrews, “An analytical model for the refractive-indexpower spectrum and its application to optical scintillation inthe atmosphere,” J. Mod. Opt. 39, 1849–1853 �1992�.

14. What is called backscatter enhancement in the Russian liter-ature arises from redirection of incident and scattered light�the former leading to target-plane scintillation� and not froman increase in the backscattering cross section. In lidarterms, it is an acceptance angle or antenna area effect and isnot related to the backscatter coefficient.

15. Yu. A. Kravtsov and A. I. Saichev, “Properties of coherentwaves reflected in a turbulent medium,” J. Opt. Soc. Am. A 2,2100–2105 �1985�.

16. Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, and A. I.Saichev, “Enhanced backscattering in optics,” in Progress inOptics, E. Wolf, ed. �Elsevier, New York, 1991�, Vol. 29, pp.67–197.

17. M. S. Belen’kii and V. L. Mironov, “Diffraction of optical radi-ation on a mirror disk in a turbulent atmosphere,” Kvant.Elektron. 5, 38–45 �1972�, cited in Ref. 16.

18. A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, “En-hanced backscattering from bodies immersed in a random ho-mogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz 16,1064–1070 �1973�, cited in Refs. 15 and 16.

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